Error estimate of a fourth-order Runge-Kutta method with only one initial derivative evaluation

In the numerical solution of differential equations it is desirable to have estimates of the local discretization (or truncation) errors of solutions at each step. The estimate may be used not only to provide some idea of the errors, but also to indicate when to adjust the step size. If the magnitude of the estimate is greater than the preassigned upper bound, the step size is reduced to achieve smaller local errors. If the magnitude of the estimate is less than the preassigned lower bound, the step size is increased to save the computing time.